Introduction to Linear Programming
The Solver Sensitivity Report explained
Week 4
MBA104 STATISTICS AND DATA ANALYSIS (Professor: Tomás Gutiérrez)
This cell shows the impact that in the objective function would have the inclusion of one unit of Jessy in our production program. In this case if we
produce one unit of Jessy, our total profit would be reduced in 1,25. Why? Because we will have to produce less of the other products that are the
ones included in the optimal solution.
Of course the Reduced Cost is only possible for variables not included in the optimal solution.
Adjustable Cells
Cell
$B$11
$C$11
$D$11
Name
Production levels Barby
Production levels Candy
Production levels Jessy
Final
Value
19
50
0
Reduced
Cost
0
0
-1,25
Objective
Coefficient
25
10
5
Allowable
Increase
55
1E+30
1,25
Allowable
Decrease
5
6,875
1E+30
Name
Gold Consumption
Diamonds Consumption
Emeralds Consumption
Labor Consumption
Polisher Consumption
Final
Value
6,25
150
50
200
114,4
Shadow
Price
0
0
6,9
3,1
0
Constraint
R.H. Side
9
160
50
200
200
Allowable
Increase
1E+30
1E+30
66,4
10
1E+30
Allowable
Decrease
2,75
10
10
150
85,6
Constraints
Cell
$E$17
$E$18
$E$19
$E$20
$E$21
The Shadow Price is the impact that an extra unit of the restriction
would have on the objective function, in this case, the profit. If we
include one more hour in the labor department, profit will go up in
3,1, It can also be understood as the maximum we should pay for
an extra unit of this resource over it's normal cost (like a premium).
Of course this only has sense when we are talking about a
restriction that is already totally consumed (bottleneck). In this
case we could try the same with the Emeralds
Would you pay for an extra unit of Emeralds 8 dollars about its
normal price? Would you pay a bonus of 3$ for each extra hour
worked in the Labor Department?
These are the values in which we can change the
coefficient of each variable in the objective function
without altering the optimal solution.
If we change the coefficient (profit per unit) of
Barby up to 80 (25+55), the optimal solution
(product mix) won't change because we have an
Allowable increase of 55.
In the same way we can decrease the profit per
unit down to 20 (25-5) without changing the optimal
solution (the allowable decrease is 5).
Of course, both cases the final profit is going to
change but you will keep the same product-mix.
The shadow price works if we inside the allowable increase and decrease of the restriction. So if
we add 10 more hours or if we reduce 150 hours. If we exceed these limits the impact on the
objective function won't be 3,1. It will be another number but we don't know it
If we choose to increase the labor deprtament in 10 hours, profit will increase in 10*3,1= 31.
If we decrease the capacity in 10 hours the benefit will do so in 1 -10*3,1= -31
How do you think the benefit will change after those limits are exceeded?
Would you pay 42,8$ for 20 extra hours in department A?
NOTE: when we increase based on the allowable increase, we do't need resources on the other
restriction, even if we have more bottlenecks. Excel is already considering those othe
bottlenecks.